Monday, October 18, 2010

MANDELBROT SET

Strange the death of the polish mathematician Benoit Mandelbrot: he died, but his ideas are more alive than ever. Mandelbrot, creator of fractal geometry (semigeométric, irregular objects repeated at different scales), understood since 1960, that clouds, turbulence, chaotic systems, behavior of prices, growth of mammals, mountains, circulatory system, snowflakes, coastlines, fluctuations in the stock market, brain tissue, the immune system, climates, etc., had similar patterns when they were studied using scales increasingly smaller (fuzzy boundaries). He understood also that he could model these phenomena with mathematical objects called fractals (fractional dimensionality).

The most famous fractal is the Mandelbrot set, a subset of the complex space, whose boundaries are fractal. In Mandelbrot set, the values for the complex number c, are built in sequence induction, so that the orbit of 0 under iteration of the complex quadratic polynomial :zn+1 = zn2 + c , remains bounded. A complex number such that if you start with z0 = 0 and applied the iteration repeatedly zn absolute value never exceeds a certain number that depends on c. In the computer the Mandelbrot set has a determined limit. As you look at the object with higher and higher resolutions, the edges of the snowman is so vague as the edges of a flame (looks like a fractional dimensionality). The predecessor of fractals is Norbert Wiener's book Cybernetics: Control and Communication in the Animal and the Machine (1948), which sought to model the performance of machines, biological phenomena, unicellular organisms, the economy of nations.